Some Recent Applications of Functional Analysis to Approximation Theory

P. L. Butzer
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P. L. Butzer: Rheinisch-Westfälische Technische Hochschule Aachen, Lehrstuhl A für Mathematik

A chapter in Zum Werk Leonhard Eulers, 1984, pp 133-155 from Springer

Abstract: Abstract A major portion of approximation theory is concerned with the approximation of functions by polynomials or by sequences {Tn} of linear operators, more specifically with the connections between the structural properties of the function f being approximated and the convergence per se and/or rate of convergence of ‖Tn (f) — f ‖ to zero for n → ∞. in particular, the wide area of approximation theory and its applications is devoted to the convergence per se and the rate of vonvergence of, for example, (a) the best trigonometric approximation of a given function, (b) the partial sums of the Fourier series of a function to the function. itself, (c) the solution of Dirichlet’s problem for the unit disk to the given boundary value, (d) the Whittaker — Shannon sampling series expansion of a duration-limited function to the functioninquestion, (e) the sums occuring in the weak law of large numbers in probability theory.

Date: 1984
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DOI: 10.1007/978-3-0348-7121-1_7

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